Adámek and Koubek [1]} showed that in a category \(\mathcal {C}\)  with limits
of all ordinal indexed chains if the final sequence of a functor \(T: \mathcal {C}\rightarrow \mathcal {C}\)  stabilizes
at ordinal \(k\) , in the sense that the arrow \(f^{k+1}_k: T^{k+1}(1) \rightarrow T^{k}(1)\)  in the final
chain is an isomorphism, then \((T^k(1),(f^{k+1}_k)^{-1})\)  is a final \(T\) -coalgebra. Moreover,
the existence of a final coalgebra is sufficient to ensure stabilization of the final sequence.
It can be shown that in \(\mathsf {SMod}\)  the final chain stabilizes at or before the index \(\omega \) .
